Heat olive oil in a deep, 12-inch non-stick skillet over medium heat. Season each chicken thigh with Cajun seasoning and black pepper. Add chicken to shimmering oil and cook until browned, about 5 minutes per side.
Add chicken broth, chopped onion, and minced garlic. Bring to a boil, cover, reduce heat to low, and simmer until the chicken is almost done, about 20 minutes. An instant-read thermometer inserted into the center should read at least 160 degrees F (71 degrees C).
Increase heat to medium, add okra, corn, and tomatoes, and bring to a boil. Replace lid, reduce heat to low, and simmer until vegetables are soft and internal temperature of the chicken is 165 degrees F (74 degrees C), about 10 more minutes.
Mix water and cornstarch together in a small bowl and stir into the sauce rapidly until thickened, about 3 minutes.